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I was reading material on hierarchical models, especially 3 level hierarchical models. One of the examples I found was having the hierarchy Hospitals-Wards-Nurses. Only 1 measurement per Nurse. So when I think of a random intercept only model for this hierarchy I was using a distribution for hospitals and then different distributions for wards in different hospitals. i.e. Effect of hospital ~ $N(0,\sigma_{hosp}^2)$ and effect of ward in hospital h are distributed as ~ $N(0, \sigma_{wards-hosp-h}^2)$. So depending on the hospital the variance of the wards is different.

I found in Multilevel Analysis by Joop J. Hox (pg. 34 to 36) that only 1 variance for all the wards is considered. i.e. effect of any ward ~ $N(0, \sigma_{ward}^2)$. I further checked SAS multilevel model primer(pg. 15) that SAS also does the same.

My question is why would we assume that irrespective of the hospital the effect of any ward is ~ $N(0, \sigma_{ward}^2)$? If I am not understanding incorrectly, then using different variances of wards per hospitals means that I am only modeling variances of wards for the hospital in the current data set. But then I also sample the variances of these wards from another distribution (I follow a Bayesian approach). So if a new hospital comes up I could give some interval estimate of what the variance of the ward effect for this hospital will be.

Am I thinking in right direction? and under what circumstances would someone do the simplification which was done in the book and SAS primer?

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Well so I ended up asking this to my Professor. And he replied back with this.

Your proposal to have the ward-level variance change from hospital to hospital is very sensible. In fact, the simpler model is often considered for convenience, and sometimes for ease of interpretation. But that does not automatically mean that the model would be fitting well to the data.

In a Bayesian setting, adding a hyper-prior for this variance, as you mention, is a sensible and, for that matter, practical way forward.

The hierarchy can be taken into account by two random statements indeed, one at the school level and one at the teacher level.

Accommodating the heterogeneity that you mention in proc mixed is not so easy; in principle, you can use the GROUP= option in the RANDOM statement, but using, for example GROUP=ward would allow a different ward variance per hospital. That sounds great, but these variances would not considered to be drawn from a distribution; in fact, there would be an entirely different variance parameter per ward. This would work only with a sufficiently small number of wards. Arguably, a Bayesian analysis is more promising.

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